26 Prof. W. Thomson on the Reduction of 



successive terms of the complex harmonic functions by which it 

 is to be expressed ; that is to say, let these constants be such 



that 



(Ft) = M + M^os \-£- - e, J + M 2 cos (-?? - e 2 J 



(Qirt 



+ M 3 cos^ r -e 3 J + &c. 



Then, expanding each cosine by the ordinary formula, and 

 assuming 



M 1 cose 1 = A l , M 2 cose 2 =A 2 , &c, 



M 1 sin6, = B 1 , M 2 sine 2 =B 2 , &c, 

 we have 



F(0 » A + A, cos -Z-t + A 2 cos -^- + A 3 cos ^- + &c, 



, „ . 2irt -r, . kirt , ^ . Qirt p 

 + B 1 8in^-+B a sin-^- + B 3 sin-^- + &c. 



Multiplying each member by cos -7^— dt, where i denotes or 

 any integer, and integrating from t = o to t — T, we have 



I F{t)cos- 7 =r-dt=PL i \ ( cos-ht- ) dt, 



Jo *- Jo 7 



= A, x ^T, when i is any integer ; 



or 

 Hence 



A = I j* T F(0 A, 



and similarly we find 



B=^l F(t)sm-Y~dt: 



equations by which the coefficients in the doable series of sines 

 and cosines are expressed in terms of the values of the function 

 supposed known from t = o to t = T. The amplitudes and 

 epochs of the single harmonic terms of the chief period and its 

 submultiples are calculated from them, according to the follow- 



